There’s no reason you can’t, and shouldn’t have both. They serve different purposes. If your newsletter/mailing list is used to announce new products, offer discounts etc. then make sure your blog never does. Your blog can be more definitive to make you the go-to expert on any subject that you are qualified in. It is more casual than a website – think of it more like the coffee shop [your blog] in a store. [your website.]
You can set up your blog to allow feedback, comments and questions from subscribers, thus making it far more interactive than a newsletter which is mainly informative. But wait! Why not use that newsletter to drive people to your blog and to subscribe? Can you use your blog to encourage people to sign up for your newsletter? Starting to get the picture here? 
Blogs are also good for search engine status and are available for the whole world to see while your newsletter is sent to people you already know. They can work well hand in hand.
Both blogs and newsletters land in inboxes and people will read either or both. This is called Marketing, folks.
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